Novel Recursive Approximation for Fractional Nonlinear Equations

ITM Web of Conferences 22, 01008 (2018) https://doi.org/10.1051/itmconf/20182201008 CMES-2018

Novel Recursive Approximation for Fractional Nonlinear Equations within Caputo-Fabrizio Operator Mehmet Yavuz1,* 1

Department of Mathematics-Computer Sciences, Necmettin Erbakan University, Konya, Turkey Abstract. This study displays a novel method for solving time-fractional nonlinear partial differential equations. The suggested method namely Laplace homotopy method (LHM) is considered with Caputo-Fabrizio fractional derivative operator. In order to show the efficiency and accuracy of the mentioned method, we have applied it to time-fractional nonlinear Klein-Gordon equation. Comparisons between obtained solutions and the exact solutions have been made and the analysis shows that recommended solution method presents a rapid convergence to the exact solutions of the problems.

1 Introduction In the last decades, fractional calculus were applied to various fields of mathematical and physical analysis such as modelling some common viruses, estimation financial processes, diffusion, relaxation processes and so on [1-11]. Especially, modelling the nonlinear problems in these areas with fractional differential equations was extensively used. In order to find out the efficiency and superiority of the fractional differential equations, a lot of studies were emerged by some scientists [12-17]. On the other hand, some important fractional derivative operators have been developed such as Caputo-Fabrizio [18] and Atangana-Baleanu [19]. These operators are very important to model the complex nonlinear fractional dynamical systems and to solve them. In recent years, some scientists have preferred these operators to other fractional operators which have a singularity [11, 20-23]. In this study, we also use the Caputo-Fabrizio fractional operator to solve nonlinear KleinGordon equation by considering the Laplace homotopy method.

2 Caputo-Fabrizio Operator and its Main Properties Definition 1. The Caputo-Fabrizio fractional derivative in Caputo sense (CFC) is defined by [18]

*

Corresponding author: [email protected]

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).


 β (t −τ )  (1)  dτ , 0 < β ≤ 1, 1− β 0 1− β  is a normalization function such that M ( 0 ) = M (1) = 1. The CF definition

CFC 0

where M ( β )

M (β )

Dtβ f ( t ) =

t

∫ f ′ (τ ) exp  −

can be applied for the functions which do not belong to H 1 ( a, b ) and the kernel does not have singularity for t = τ unlike the Caputo fractional derivative. Definition 2. The Laplace transform (LT) of

L

{

CFC 0

}

Dt( β + n ) f ( t ) ( s ) = =

CFC 0

Dtβ f ( t ) is given by [18, 24]:

  β t  1 L f ( β + n ) ( t ) L exp  −  1− β  1− β  

{

}

s n +1 L { f ( t )} − s n f ( 0 ) − s n −1 f ′ ( 0 ) − ⋯ − f ( n ) ( 0 ) s + β (1 − s )

(2)

.

Definition 3. Let 0 < β < 1. The CFC fractional integral of order β is defined by [25, 26]: CFC 0

I β f (t ) =

2 (1 − β )

(2 − β ) M (β )

f (t ) +

2β f (τ ) dτ , t ≥ 0. ( 2 − β ) M ( β ) ∫0 t

(3)

3 Role of the Suggested Fractional Operator in the Fundamental Method Consider the following nonlinear FPDE: CFC Dtβ φ ( x, t ) + µ (φ ( x, t ) ) + ω (φ ( x, t ) ) = ν ( x, t ) , with initial condition where

CFC

β

Dt

( x, t ) ∈ [ 0,1] × [0, T ] ,

φ ( x, 0 ) = k ( x ) ,

0 < β ≤ 1,

(4) (5)

shows the Caputo-Fabrizio-Caputo fractional derivative of order β ,

ω (φ ( x, t ) ) is a non-linear operator and ν ( x, t ) is a known function and φ ( x, t ) is an

unknown function. Using the LT of the CF operator we define L {φ ( x, t )} ( s ) = ∆ ( x, s ) .

Then we have the following homotopies:  β + s (1 − β )  1 ∆ ( x, s ) =  (6)  νɶ ( x, s ) − L µ (φ ( x, t ) ) + ω (φ ( x, t ) )  + (φ0 ( x ) ) . s s   where ∆ ( x, s ) = L {φ ( x, t )} and νɶ ( x, s ) = L {ν ( x, t )} . Applying the suggested method we

{

}

obtain the approximate solution as

∞

∆ ( x, s ) = ∑ p m ∆ m ( x, s ), m = 0,1, 2,...,

(7)

m=0

where ∆ m ( x, s ) values are known functions. The nonlinear term ω (φ ( x, t ) ) in Eq. (4) can be evaluate as ∞

ω (φ ( x, t ) ) = ∑ p mη m ( x, t ), m=0

and the polynomials η m ( x, t ) are given in [27] as

2

(8)


  ∞ i  (9) ω  ∑ p φi   , m = 0,1, 2,….   p =0   i=0 Substituting Eqs.(8) and (9) into Eq.(6) we construct the solution series for the main problem. ∞ ∞  β + s (1 − β )   ∞ m   m p m ∆ m ( x, s ) = − p    L  ∑ p ∆ m ( x , t ) + ∑ p η m ( x, t )    ∑ s m=0 m=0      m =0  (10)  β + s (1 − β )  1 + (φ0 ( x ) ) +  νɶ ( x, s ) . s s   By equating the same powers of p, we construct the homotopy steps of as

η m (φ0 , φ1 ,..., φm ) =

p 0 : ∆ 0 ( x, s ) =

1 ∂m m ! ∂p m

 β + s (1 − β )  1 φ0 ( x ) ) +  ( νɶ ( x, s ) , s s  

 β + s (1 − β )  p1 : ∆1 ( x, s ) = −   L µ (φ0 ( x, t ) ) + ω (φ0 ( x, t ) ) , s    β + s (1 − β )  p 2 : ∆ 2 ( x, s ) = −   L µ (φ1 ( x, t ) ) + ω (φ1 ( x, t ) ) , s   ⋮

{

}

{

}

(11)

 β + s (1 − β )  p n +1 : ∆ n +1 ( x, s ) = −   L µ (φn ( x, t ) ) + ω (φn ( x, t ) ) . s  

{

}

Considering p → 1, we can see Eq.(11) gives the approximate solution for problem (10) and the solution is given by n

ℑn ( x, s ) = ∑ ∆ j ( x, s ).

(12)

j =0

Taking the inverse Laplace transform of the sum (12), we have the approximate solution

{

}

φapprox ( x, t ) ≅ φn ( x, t ) = L−1 ℑn ( x, s ) .

4 Solution of Nonlinear Fractional Klein-Gordon Equation Consider the fractional nonlinear homogeneous Klein-Gordon equation (KGE) [28] CFC Dtβ φ ( x, t ) − φxx ( x, t ) + φ 2 ( x, t ) = 0, x ∈ R, t > 0, 0 < β ≤ 1, subject to the initial condition

φ ( x, 0 ) = 1 + sin x,

(13)

(14) x ∈ R. Firstly, applying the mentioned method in Section 3, to Eq.(13) with the initial condition (14), we obtain the approximate solution as  β + s (1 − β )  1 2 ∆ ( x, s ) =  (15)  L {φxx ( x, t ) − φ ( x, t )} + φ ( x, 0 )  . s s   Then, we consider the inverse Laplace transform of Eq.(15), we have  β + s (1 − β )   2 φ ( x, t ) = φ ( x, 0 ) + L−1  (16)  L {φxx ( x, t ) − φ ( x, t )} . s   

3


The nonlinear polynomials η m ( x, t ) are given in [27] as

η0 (φ ) = (1 + sin x ) , 2

η1 (φ ) = −2 (1 + sin x ) ( sin 2 x + 3sin x + 1) ( β t − β + 1) ,

(

)

η 2 (φ ) = ( sin 2 x + 3sin x + 1) ( β t − β + 1) + 2 (1 + sin x )( 3sin x − 2 cos 2 x )( β t − β + 1) (17) 2

  β 2t 2 2 + 4 (1 + sin x ) ( sin 2 x + 3sin x + 1)  β 2 − 2 β + 1 + − 2(β 2 − β )t , 2   ⋮ Moreover, we use the series solution in Eq.(10), we can write ∞ ∞  β + s (1 − β )   ∞ m  1 m p m ∆ m ( x, s ) = − p   L  ∑ p ∆ m ( x, t ) + ∑ p η m ( x, t )  + (φ0 ( x ) ) . (18) ∑ s m=0 m=0  s   m =0 By comparing the coefficients of powers of p, we construct the homotopy steps as in Eq.(11) and taking the inverse LT of obtained values, we have the iteration steps as φ0 ( x, t ) = 1 + sin x,

φ1 ( x, t ) = − ( sin 2 x + 3sin x + 1) ( β t − β + 1) , 

φ2 ( x, t ) = 2 (1 + sin x ) ( sin 2 x + 3sin x + 1)  β 2 − 2β + 1 +  + ( 3sin x − 2 cos 2 x )( β t − β + 1) ,

β 2t 2

 − 2(β 2 − β )t  2 

(19)

⋮  β + s (1 − β )    L µ (φn ( x, t ) ) + ω (φn ( x, t ) )  . s    Finally, considering the sum (19), we have the approximate solution φapprox ( x, t ) ≅ φn ( x, t ) = 1 + sin x − ( sin 2 x + 2 cos 2 x + 1) ( β t − β + 1)

φn +1 ( x, t ) = L−1 

{

}

  β 2t 2 + 2 (1 + sin x ) ( sin 2 x + 3sin x + 1)  β 2 − 2β + 1 + − 2 ( β 2 − β ) t  + …. 2   Following figures show the solution functions for some different values of x, t and fractional parameter β .

Fig.1. Solutions of Eq.(13) for β = 1 (left) β = 0.85 (right)

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Fig.2. Solutions of Eq.(13) for β = 0.5 (left) β = 0.1 (right)

Fig.3. Solutions of Eq.(13) for x = 0.1 (left) x = 1 (right)

4 Conclusions In this study, Laplace homotopy method which is defined with a new fractional operator is considered. This method presents an efficient solution for nonlinear FDEs without discretization. Therefore, this method can be applied to nonlinear FDEs easily. The nonlinear terms in the equation are evaluated by using Adomian polynomials. In this context, we obtain the approximate-analytical solution of nonlinear KGE of fractional order by using the LHM. The results obtained in this study carried out that the method is very efficient and accurate method for solving the fractional nonlinear homogeneous KGE. Maple software has been used for obtaining the numerical computations and presenting the solution plots with respect to various values of the variables.

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